Does the Budget Close? Radiated Energy, Final Spin, and the Consistency of the Remnant Description in Three Mergers

A merger's remnant is described by numbers that must satisfy an accounting identity: the initial mass minus the radiated energy must equal the remnant mass, and the remnant spin must be consistent with the angular momentum the progenitors brought in minus what the waves carried away. This paper checks that accounting for three published mergers using catalogue values alone, and reports the residuals. For GW150914 the budget reads 65.0 solar masses in total, minus 3.0 radiated, against a published remnant of 62.0, leaving a residual of 0.00 solar masses, or 0.00 per cent. Across the three systems the largest absolute residual is 0.93 per cent of the total mass, from GW190521, a level consistent with the rounding of the published values and with no systematic direction. The published remnant spins all lie inside the interval bounded by the no-loss and maximal-loss remnant-fit values, so the angular-momentum side of the accounting raises no exception either. The mass budget closes to within 0.93 per cent across three mergers, with the GW150914 residual at 0.00 per cent, and all published spins lie inside their allowed accounting intervals. This study is computed from stated assumptions and published catalogue values; it is simulated, not measured.